Universality conjecture and results for a model of several coupled positive-definite matrices

The paper contains two main parts: in the first part, we analyze the general case of $p\geq 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a $(p+1)\times (p+1)$ matrix valued solution of a Riemann-Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields which include the Bessel$_\nu$ universality for $1$-level and the Meijer-$G$ universality for $2$-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a $(p+1)\times (p+1)$ origin parametrix in terms of Meijer G-functions. The solution of the full Riemann-Hilbert problem is derived rigorously only for $p=3$ but the general framework of the proof can be extended to the Cauchy chain of arbitrary length $p$.Comment: 45 pages, 7 figures. To appear in Commun. Math. Phys. Version 2 contains minor changes in the introduction and updates literatur

Isomonodromic deformation of resonant rational connections

We analyze isomonodromic deformations of rational connections on the Riemann sphere with Fuchsian and irregular singularities. The Fuchsian singularities are allowed to be of arbitrary resonant index; the irregular singularities are also allowed to be resonant in the sense that the leading coefficient matrix at each singularity may have arbitrary Jordan canonical form, with a genericity condition on the Lidskii submatrix of the subleading term. We also give the relevant notion of isomonodromic tau function extending the one of non-resonant deformations introduced by Miwa-Jimbo-Ueno. The tau function is expressed purely in terms of spectral invariants of the matrix of the connection.Comment: 48 page